Hi everyone this is the 3rd video in our chapter
straight lines. Today we will be discussing the angle between 2 straight lines and 2 special
occasions of this angle. Lets take these 2 lines l1 and l2. Lets say
that the gradient of this line is m1 and this line m2. We can say that m1=tan(theta) and
m2=tan(beta). Let alpha be the angle between these 2 lines. So now our goal is to find
an equation for this angle ‘alpha’ Since this is a triangle we know that the
exterior angle is equal to the sum of the opposite interior angles.
So beta=alpha + theta. If we subject alpha we get alpha=beta – theta. We can also say
that tan(alpha)=tan (beta – theta). From trigonometry by using this equation tan(A
– B)=tan A – tan B/(1 + tan A*tan B), we can expand the right hand side. This gives us tan(beta) – tan(theta)/1 + tan(beta)*tan(theta).
If we substitute the gradients into this equation, we get
tan(alpha)=m2 – m1/(1 + m1m2). Notice that there are 2 angles between any
2 lines. An acute angle and an obtuse angle. We can see here that this angle ‘gamma’ is
180 – alpha. So tan(gamma)=tan(180 – alpha), so this will give us ‘-tan(alpha)’. So that
will be (m2 m1)/(1 + m1*m2). So what this means that is that if we get the value for
tan(alpha) is positive, then we have the acute angle and if the value is negative, then we
have the obtuse angle. To consider both of these angles, we put this equation in a modulus.
So this is the final equation, tan(alpha)=’modulus’ (m2 – m1)/(1 + m1*m2)
So now we can find the angle between 2 lines in terms of their gradients. Now for the tip of the day
Notice that when alpha decreases more and more and becomes zero, the 2 lines become
parallel to each other. So now we can say that tan 0=0, so (m2 – m1)/(1
+ m1*m2)=0. So then we get m1=m2. So when 2 lines are parallel the gradients are equal
to each other. Similarly when alpha equals 90 degrees, then
tan(90)=infinity. So the denominator is 0. So this gives (m2 – m1)/(1 + m1*m2)=1/0
since it is infinity. So now we can find that m1*m2=-1. So this means when the lines are
perpendicular to each other then the product of their gradients equals -1. That brings us to the end of this video
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